Chapter 16 Probability
Activity
• Rock-Paper-Scissors Shoot Tournament
1) Pair up and choose one person to be person
A and the other person B.
2) Play 9 games of RPS keeping track of the wins
of person A vs wins of person B vs ties (a tie
counts as a game, only play more than 9
games if you have to break a tie).
3) Top 16 winners advance….
Experimental Probability
• Lets collect and analyze the data to see if rockpaper-scissors is a fair game.
Theoretical Probability
• Let’s create a tree diagram to analyze if rockpaper-scissors is a fair game.
Is 3-way RPS fair?
• Lets analyze this with theoretical probability.
Rules: Person A wins if all three signs are the
same, person B wins if two signs are the same
and person C wins if all three signs are different.
Definitions
• Sample Space
The sample space, S, of a random
phenomenon is the set of all possible
outcomes.
• Event
An event is a set of outcomes of a random
phenomenon; that is, a subset of the sample
space.
Definitions
• Trial
A trial is a single occurrence of the random
phenomenon.
A single trial can result in any one of the
possible outcomes in the sample space.
• What was the sample space, event and trial in
our three person RPS example?
• Sample Space: {RRR, PPP, SSS, RRP, RPR, PRR,
RRS, RSR, SRR, PPR, PRP, RPP, PPS, PSP, SPP,
SSR, SRS, RSS, SSP, SPS, PSS, RSP}
• A trial could be any of the 18 outcomes in the
sample space.
• Examples of Events: Player 1 winning, Player 2
winning, Player 3 winning, Getting Paper.
Sample Space: {RRR, PPP, SSS, RRP, RPR, PRR,
RRS, RSR, SRR, PPR, PRP, RPP, PPS, PSP, SPP, SSR,
SRS, RSS, SSP, SPS, PSS, RSP}
Find the probability of each Event:
Player 1 winning
Player 2 winning
Player 3 winning
Getting Paper
Example 1
• If the random phenomenon is drawing a card
at random out of a standard deck of cards,
then the sample space is the set of individual
cards in the deck:
S = {A♠, 2♠, . . . , Q♠, K♠,
A♣, 2♣, . . . , Q♣, K♣,
A♥, 2♥, . . . , Q♥, K♥,
A♦, 2♦, . . . , Q♦, K♦}
There are 52 possible outcomes in this sample
space.
Example 1 (cont’d)
• If an event, call it A, is drawing a card at random
and getting a queen, then the event is:
A = {Q♠, Q♣, Q♥, Q♦}
There are 4 possible outcomes in this event.
• A trial would be drawing one card at random
from the deck one time.
Example 1.
• If event A is drawing a card at random from
a standard deck of cards and getting a
queen, then:
P(A)=
Count of outcomes in A
Count of outcomes in S
4
1


52 13
Example 2
• If the random phenomenon is tossing a fair
coin twice, then the sample space is the set of
all possible outcomes of the coin toss:
S = { HH, HT, TH, TT }
There are 4 possible outcomes in this sample
space.
Example 2 (cont’d)
• If an event, call it B, is tossing a fair coin twice
and getting at least one Tails, then the event is:
B = { HT, TH, TT }
There are 3 possible outcomes in this event.
• A trial would be tossing a single coin two times.
Example 2.
• If event B is tossing a fair coin twice and
getting at least one tails, then:
P(B) =
Count of outcomes in B
Count of outcomes in S
3

4
Notation
Notation
P(A)
Read As
Means
“P of A”
The probability that
event A will occur on
any given trial of the
random phenomenon
Basic Probability Rules
• Rule # 1: 0 ≤ P(A) ≤ 1
The probability of any event A that may occur
is a number between 0 and 1.
0 < P(A) < 1
An event with P(A) = 0 never occurs.
An event with P(A) = 1 always occurs.
Basic Probability Rules
• Rule # 2: P(S) = 1
The collection S of all possible outcomes has
probability 1.
In other words, every trial must result in one
of the possible outcomes.
Guided Practice
You roll a blue and red die at the same time.
1. List the sample space.
Guided Practice
You roll a blue and red die at the same time.
2. List the event that you get doubles.
Guided Practice
You roll a blue and red die at the same time.
3. Find the following probabilities:
a. not getting doubles.
Guided Practice
You roll a blue and red die at the same time.
3. Find the following probabilities:
b. getting a 5 on the blue die.
Guided Practice
You roll a blue and red die at the same time.
3. Find the following probabilities:
c. getting a 5 on at least one die.
Complement Rule
• The probability that the complement of an
event occurs is equal to one minus the
probability that the event does occur.
P(A′) = 1 – P(A)
Example 1.
• If event A is drawing a card at random from
a standard deck of cards and getting a
queen, then the probability of not drawing a
queen is:
P ( A' )  1  P ( A)
1
 1
13
12

13
Example 2.
• If event B is tossing a fair coin twice and
getting at least one tails, then the
probability of not getting a tails on either
toss is:
P( B' )  1  P( B)
3
 1
4
1

4
Definitions
• Disjoint Events
Events that have no common outcomes are
called disjoint events.
• Union of Events (“OR”)
The union of two events is the combined set
of outcomes in those events. If the same
outcome is in both events, it is only listed
once in the union set.
Notation
Notation
AB
Read As
Means
“A union B”
or
“A or B”
The event that
either A or B
occurs
(note: both A
and B could
occur)
Addition Rule for Disjoint Events
• If events A and B are disjoint, then
P(A or B) = P(A) + P(B)
Set notation:
P(A  B) = P(A) + P(B)
Guided Practice
You roll a die and toss a coin.
1. List the sample space.
2. Let A be the event that you get an even
number on the die and tails on the coin.
Let B be the event that you get heads on the
coin. List A and B.
3. a. Find P(A), P(B) and P(A or B)
b. Are events A and B disjoint?
Notation
Notation
Read As
Means
AB
“A intersect B”
or
“A and B”
The event that
both A and B
occur
General Addition Rule
P(A or B) = P(A) + P(B) – P(A and B)
Set notation:
P(A  B) = P(A) + P(B) – P(A  B)
Practice 3.
You roll a die and toss a coin.
1. Let A be the event that you get an even
number on the die.
Let B be the event that you get tails on the
coin. List A and B.
2. a. Find P(A), P(B), P(A or B), P(A and B)
b. Are events A and B disjoint?
Tree Diagrams
• Each set of branches from one starting point must
equal 1.
• Multiply the probabilities along each branch to get
the probability of all the events on that branch
occurring.
• The sum of all the final probabilities
must equal 1.
 Useful for computing probabilities of
independent observations.
Example:
What is the probability that you will roll a die
three times and get at least two threes?
Draw a tree diagram to illustrate this.
1/6
1/6
3
3
5/6
Not
3
Begin
5/6
Not
3
1/6
3
5/6
Not
3
1/6
3
5/6
1/6
Not 3
5/6
Not 3
1/6
5/6
1/6
5/6
3
3
1/216
5/216
5/216
25/216
5/216
Not 3 25/216
3
25/216
Not 3 125/216
Probability Distribution
• List all the possible outcomes.
• List the probability of each outcome.
• The sum of all the probabilities must = 1.
 Useful for computing probabilities of
events that are not equally likely.
Example:
You roll two dice and add the values. Complete
a probability distribution for the sum of the two
dice.
Step 1: Write out the possible outcomes.
Step 2: Calculate the probability of each outcome.
Step 3: Summarize the probability distribution.
Example:
Second Die Value
First
Die
Value
Sums
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Practice 4.
A couple has three children. Given that the
probability of any child born a boy is ½,
a. Draw a tree diagram to illustrate the possible
genders of the three children, in order.
b. What is the probability that the couple has
exactly two girls?
c. Complete a probability distribution for the
number of girls the couple could have.